Good ridge estimators based on prior information

Ridge regression is re-examined and ridge estimators based on prior information are introduced. A necessary and sufficient condition is given for such ridge estimators to yield estimators of every nonnull linear combination of the regression coefficients with smaller mean square error than that of the Gauss-Markov best linear unbiased estimator.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

Superiority comparisons between mixed regression estimators

This paper gives necessary and sufficient conditions for a mixed regression estimator to be superior to another mixed estimator. The comparisons are based on the mean square error matrices of the estimators. Both estimators are allowed to be biased.     

A relationship between generalized and integrated mean square errors

Generalised Mean squared error is a flexible measure of the adequancy of ? repression estimator. It allows specific characteristics of the regression model and its intended use to be In-corportated in the measure itself. Similarly, integrated mean squared error enables a researcher to stipulate particular regions of interest and wi ighting functions in the assessment of a prediction equation. The appeal of both measures is their ability to allow design or model characteristics to directly influence the evaluation of fitted regression models. In this note an e-quivalence of the two measures is established for correctly specified models.     

A simulation study of the stochastic ridge k

A simulation of regressions is used to generate estimates by iteration of the generalized ridge parameter.The simulation results indicate that generalized ridge parameters estimated by iteration from near collinear data may be quite different from unknown optimal values.     

Equivariance of ridge estimators through standardization,a note

We show that standardization of data recommended for Ridge Regression (RR) by Hoerl and Kennard (1970 a8b), Marquardt and Snee (1975), and others is worthwhile despite the criticisms by Swamy, Mehta and Rappoport (SMR) (1978).     

A note on some conditions for optimality of certain ridge estimators

Consider the linear regression model, y = Xβ + ε in the usual notation with X'X being in the correlation form. Galpin(1980) claimed that the ridge estimators of Hoerl, Kennard and Baldwin(1975) and Lawless and Wang(1976) give guaranteed lower mean squared error than the least squares estimator when X'X has at least two very small eigen values. We show that the arguments of Galpin(1980) leading to the above claim are incorrect, and hence the claim itself is unsubstantited. A Monte Carlo study shows that Galpin's claim is not correct in general.     

Comparisons among regression estimators under the generalized mean square error criterion

It is not always prossible to establish a preference ordering among regression estimators in terms of the generalized mean square error criterion. In the paper, we determine when it is feasible to use this criteion to couduct comparisons among ordinary least squares, principal components, ridge regression, and shrunken least squares estimators.     

Some finite sample properties of generalized ridge regression estimators

In this paper we study the mean square error properties of the generalized ridge estimator. We obtain the exact and the approximate bias and the mean square error of the operational generalized ridge estimator in terms of G( ) functions. We show, among other things, that the operational generalized ridge estimator does not dominate the ordinary least squares estimator up to a certain order of approximation. Finally, we note that the iterative procedures to obtain coverging ridge estimators should be used with caution.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Superiority comparisons of homogeneous linear estimators

In the paper homogeneous linear estimators of the parameter vector of the general linear model are compared in terms of their MSE matrices. A necessary and sufficient condition for the difference of two MSE matrices to be positive definite is obtained and its practical existence discussed. The non-negative definiteness of the difference also receives attention, and conditions for this case are discussed. The absence of any conditions of the above type is taken into consideration as well.     

On biased linear estimators in models with arbitrary rank

In this paper we define a class of biased linear estimators for the unknown parameters in linear models with arbitrary rank. The feature of our approach is to reduce the estimation problem in arbitrary rank models to the one in full-rank models. Some important properties are discussed. As special cases of our class, we extend to deficient-rank models six known biased linear estimators.     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.     

Combining the unrestricted estimators into a single estimator and a simulation study on the unrestricted estimators

The purpose of this paper is to combine several regression estimators (ordinary least squares (OLS), ridge, contraction, principal components regression (PCR), Liu, r?k and r?d class estimators) into a single estimator. The conditions for the superiority of this new estimator over the PCR, the r?k class, the r?d class, β?(k, d), OLS, ridge, Liu and contraction estimators are derived by the scalar mean square error criterion and the estimators of the biasing parameters for this new estimator are examined. Also, a numerical example based on Hald data and a simulation study are used to illustrate the results.     

A Note on Superiority Comparisons of Homogeneous Linear Estimators

This note extends some results on homogeneous linear estimators to the general, even nonlinear case.A Sufficient condition for the difference of mean square error matrices of minimum conditional mean square error estimator and minimum average risk linear estimator to be postive definite is derived.     

A note on combining ridge and principal component regression

Baye and Parker (1984) proposed the r-k class estimator. The purpose of this note is to deal with the comparisons among the r-k class estimators in terms of the mean square error criterion.     

Some properties of generalized ridge estimators

Exact expressions, in the form of infinite series expansions, are given for the first and second moments of two well known generalized ridge estimators. These series expansions are then evaluated using recursive formulas and computations are verified using approximations. Results are presented for the relative mean square error and bias of these estimators as well as their relative efficiency with respect to least squares.     

The feasible generalized restricted ridge regression estimator

The presence of autocorrelation in errors and multicollinearity among the regressors have undesirable effects on the least-squares regression. There are a wide range of methods which are proposed to overcome the usefulness of the ordinary least-squares estimator or the generalized least-squares estimator, such as the Stein-rule, restricted least-squares or ridge estimator. Therefore, we introduce a new feasible generalized restricted ridge regression (FGRR) estimator to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model. We also derive some statistical properties of the FGRR estimator and comparisons have been conducted using matrix mean-square error. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.     

A note on minimum average risk estimators for coefficients in linear models

In this note, we have derived a set of necessary and sufficient conditions for the biased estimators analyzed by Swamy and Mehta (1976) to be better than the generalized least squares estimator of the coefficient vector in a standard linear regression model.     

Operational ridge regression estimators under the prediction goal

In this paper we study the Mean Square Error and Conditional Mean Forecasting of Operational Ordinary Ridge Estimator. We use the G( ) functions to provide both the exact and the approximate bias and Mean Square Error of ordinary ridge estimator (ORE), We show, among other things, that ORE dominates OLS up to a certain order of approximation under the conditional mean forecasting sense.